Heriot-Watt University Research Gateway L-Algebra Models and Higher Chern-Simons Theories Citation for published version: Ritter, P & Saemann, C 2016, 'L-Algebra Models and Higher Chern-Simons Theories', Reviews in Mathematical Physics, vol. 28, no. 9, 1650021. https://doi.org/10.1142/S0129055X16500215 Digital Object Identifier (DOI): 10.1142/S0129055X16500215 Link: Link to publication record in Heriot-Watt Research Portal Document Version: Peer reviewed version Published In: Reviews in Mathematical Physics Publisher Rights Statement: Electronic version of an article published as Rev. Math. 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Oct. 2021 DIFA 2015 EMPG{15{17 L1-Algebra Models and Higher Chern-Simons Theories Patricia Rittera and Christian S¨amannb a Dipartimento di Fisica ed Astronomia Universit`adi Bologna and INFN, Sezione di Bologna Via Irnerio 46, I-40126 Bologna, Italy b Maxwell Institute for Mathematical Sciences Department of Mathematics, Heriot-Watt University Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. Email: [email protected] , [email protected] Abstract We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In a first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theo- ries form a universal starting point for the construction of L1-algebra models. We then show how to describe superconformal field theories and how to per- form dimensional reductions in this context. In a second part, we demonstrate arXiv:1511.08201v3 [hep-th] 10 Oct 2016 that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie p- algebra extensions of so(p + 2). Finally, we study a number of L1-algebra models which are physically interesting and which exhibit quantized multisym- plectic manifolds as vacuum solutions. Contents 1 Introduction and results1 2 Metric Lie p-algebroids and symplectic NQ-manifolds4 2.1 NQ-manifolds . .4 2.2 P-manifolds and symplectic NQ-manifolds . .6 2.3 L1-algebras and L1-algebroids . .8 2.4 Metric L1-algebroids . .9 2.5 Weil algebras and invariant polynomials . 11 3 Higher gauge theory with symplectic NQ-manifolds 12 3.1 Motivating example: Ordinary gauge theory . 13 3.2 General higher gauge theory . 14 3.3 Fake curvatures and dga-morphisms . 16 3.4 Higher Chern-Simons actions and AKSZ σ-models . 16 4 Supersymmetric gauge theory with NQ-manifolds 19 4.1 Maximally supersymmetric field theories . 19 4.2 Dimensional reduction . 21 4.3 Supersymmetric gauge theories using NQ-manifolds . 23 5 Nambu-Poisson and multisymplectic manifolds 24 5.1 The IKKT model and emergent geometry . 24 5.2 Nambu-Poisson structures . 25 5.3 Heisenberg Lie p-algebra . 27 5.4 Multisymplectic manifolds . 28 5.5 Quantization and emergence of multisymplectic manifolds . 30 5.6 Central Lie p-algebra extensions of so(n + 1) . 31 6 L1-algebra models 32 6.1 Homogeneous L1-algebra models of Yang-Mills type . 32 6.2 L1-algebra model of the (2; 0)-theory . 33 6.3 L1-algebra models from higher Chern-Simons theory . 34 6.4 Homotopy Maurer-Cartan equations and physically motivated models . 35 6.5 Multisymplectic solutions to L1-algebra Chern-Simons theory . 36 1. Introduction and results Most of our current physical theories, including those describing gravity and string theory, are not truly background independent. That is, their setup includes a given spacetime topology or even a detailed spacetime geometry. It is clear, however, that both topology and geometry of spacetime should ideally emerge from a more fundamental description [1]. There are a number of approaches to solve this problem, but the one which is closest 1 to what we have in mind here is that of the IKKT matrix model [2], see also [3]. This model is equivalent to ten-dimensional super Yang-Mills theory dimensionally reduced to a point and its degrees of freedom are described in a supermultiplet consisting of ten Lie algebra valued matrices and their 16 fermionic superpartners. It has been conjectured that the IKKT model provides a non-perturbative definition of type IIB superstring theory. Interestingly, vacuum solutions of this model and its deformations by background fluxes can be interpreted as noncommutative spaces. Moreover, the physics of small fluctuations around these vacuum solutions corresponds to noncommutative gauge theories. Even the dynamics of the noncommutative spaces themselves can be extracted and related to a theory of noncommutative gravity, see [4,5] and references therein. While this picture is very appealing, there are a number of reasons to consider gener- alizations in terms of higher Lie algebras or, equivalently, L1-algebras: First and from a mathematician's perspective, it is always a good idea to study deformations and categori- fications of mathematical objects to gain a more complete understanding of them. From a string theorist's perspective, there are many situations in which ordinary Lie algebras are not sufficient, but have to be replaced by categorified Lie algebras. These range from effective descriptions of M-theory over Kontsevich's deformation quantization to string field theory. For a more comprehensive list with more details and references, we refer to the introduction of [6]. From a quantum field theorists perspective, L1-algebras appear very naturally in BV- quantization of classical field theories. Moreover, L1-algebras come with a canonical equa- tion of motion, the homotopy Maurer-Cartan equation. This equation is somewhat univer- sal in the sense that many other physical field theories such as Yang-Mills theories and the recently popular M2-brane models of [7,8] can be written as a homotopy Maurer-Cartan equation on some L1-algebra, cf. e.g. [9, 10, 11]. From a gravitational physicist's perspective, there is evidence that the version of non- commutative gravity obtained from the IKKT model is too restrictive [12, 13]. This is perhaps not very surprising, as the noncommutative spaces arising in the IKKT model are K¨ahlermanifolds, and a restriction to those is clearly insufficient. As suggested in [12, 13], one should turn to Nambu-Poisson manifolds, which have enough overlap with multisymplectic manifolds as we shall explain in detail. On multisymplectic manifolds, the Poisson algebra is replaced by a higher Lie algebra of observables [14]. Again, we are led to generalizing the IKKT model to a model involving L1-algebras. It is the latter point that provided most of our motivation to continue the study of models built from categorified Lie algebras we began in [6], considering models employ- ing 2-term L1-algebras. In the present paper, we generalize the discussion to arbitrary (truncated) L1-algebras and refer to the resulting models as L1-algebra models. One of the most important problems to be solved here is that of selecting a physically interest- ing action. Beyond the dimensional reduction of the bosonic part of the six-dimensional (2,0)-theory, which we will discuss in section 6.2, there is no obvious candidate. Since other versions of higher Yang-Mills theory are unknown, we focus on higher versions of Chern-Simons theories. These theories are conveniently constructed using the language of 2 the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) formalism [15], which we review to the necessary extent. In this construction, both spacetime and higher gauge algebroid are regarded as NQ-manifolds and a connection is described as a morphism of graded manifolds between them, cf. [16, 17, 18]. The curvature then measures the failure of this morphism to be a morphism of NQ-manifold. Examples of NQ-manifolds include Poisson and symplectic manifolds as well as Courant algebroids [19]. We then show how this description generalizes to supersymmetric field theories by employing NQ-supermanifolds. In particular, we show how to describe maximally super- symmetric Yang-Mills theory and a recently proposed version of the superconformal field theory in six dimensions [20, 21, 22]. This is done by encoding the corresponding super- multiplets into a connection on superspace, which is flat along certain subspaces of this superspace. In order to follow analogous routes to the construction of the IKKT model, we also study the dimensional reduction of NQ-manifolds and its effect on connections encoded in morphisms between them. While dimensional reduction was discussed previously to some degree in [23], see also [24] for the case of Courant algebroids, we provide a more complete picture for arbitrary NQ-manifolds here. Applying this reduction procedure to the higher Chern-Simons theories obtained previously then yields an interesting set of examples of L1-algebra models. In the second part of this paper, we study solutions of L1-algebra models and connect them to certain higher quantum geometries. Just as quantizations of certain symplectic manifolds provide examples of solutions to the IKKT model, we expect higher quantiza- tions of certain multisymplectic manifolds to provide classical solutions of the L1-algebra models.